D₆₉₇₂ #72212463

order	13944
automorphism	Aut(G) order 13720896
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₉₇₂

Splittings

Fitting	C₃×C₄×C₇×C₈₃ ⋊ C₂
Hall	C₃×C₇×C₈₃ ⋊ D₄
Hall	C₇×C₈₃ ⋊ D₁₂
Hall	C₃×C₈₃ ⋊ D₂₈
Sylow	C₈₃ ⋊ D₈₄
Hall	C₃×C₇ ⋊ D₃₃₂
Sylow	C₇ ⋊ D₉₉₆
Sylow	C₃ ⋊ D₂₃₂₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₇₄₃
commutator, socle, cosocle	C₂×C₃×C₇×C₈₃	C₂²
Fitting	C₃×C₄×C₇×C₈₃	C₂
hypercenter	C₄	D₁₇₄₃
hypocenter	C₃×C₇×C₈₃	D₄
LCS	G ▷ C₂×C₃×C₇×C₈₃ ▷ C₃×C₇×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 6972

A minimal faithful permutation representation — 2 generators on 97 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(2,4)(6,7)(9,14)(10,13)(11,12)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^3,3,7,83|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^82

Conjugacy classes & character table

3489 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₉₄₄, D₂₀₉₁₆
maximal normal subgroup of	D₁₃₉₄₄
maximal quotient of	D₁₃₉₄₄, Dic₆₉₇₂, D₂₀₉₁₆

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 83]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6972 #72212463